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Original Paper ________________________________________________________ Forma, 19, 405–412, 2004

Sequences of Polyomino and Polyhex Graphs whose PerfectMatching Numbers are Fibonacci or Lucas Numbers: The Golden

Family Graphs of a New Category

Haruo HOSOYA

Ochanomizu University, Bunkyo-ku, Tokyo 112-8610, JapanE-mail address: hosoya@is.ocha.ac.jp

(Received February 15, 2005; Accepted March 15, 2005)

Keywords: Perfect Matching, Kekulé Structure, Fibonacci Numbers, Lucas Numbers,Polyomino, Polyhex

Abstract. Several sequences of graphs are introduced whose perfect matching numbers,or Kekulé numbers, K(G), are either Fibonacci or Lucas numbers, or their multiples. Sincethe ratio of the K(G)s of consecutive members converges to the golden ratio, thesesequences of graphs belong to another class of golden family graphs.

1. Introduction

A graph, G, is a mathematical object composed of vertices, {V}, and edges {E}, wherean edge spans a pair of vertices (HARARY, 1969). A matching of a graph is a set of edgesof G such that no two of them share a vertex in common. If a graph with even n = |V| hasa matching with n/2 edges it is called a perfect matching graph. The number of possibleperfect matchings of G is the perfect matching number, or Kekulé number, K(G). Althougha tree graph has at most one Kekulé structure, a number of interesting features have beenfound for the K(G) numbers of polycyclic graphs (HOSOYA, 1986). In chemistry the K(G)of a polyhex graph, or hexagonal animal, reflects the stability of the parent polycyclicaromatic hydrocarbon molecule, such as, naphthalene or benzopyrene (CYVIN and GUTMAN,1988; TRINAJSTIC, 1992). On the other hand, in solid state physics K(G) enumeration forgiant polyominoes, or tetragonal animals, is an important subject for discussing themagnetic properties of metals and antiferromagnetic substances (KASTELEYN, 1967:TEMPERLEY, 1981). A variety of useful and interesting methods for enumerating K(G) ofpolyhex and polyomino graphs are proposed and discussed.

The present author has accumulated data of the K(G) numbers of both polyhexes(YAMAGUCHI et al., 1975; HOSOYA et al., 1986) and polyominoes (MOTOYAMA andHOSOYA, 1976), and he also has proposed several mathematical techniques (HOSOYA andYAMAGUCHI, 1976; MOTOYAMA and HOSOYA, 1977; HOSOYA and OHKAMI, 1983) usefulfor the study of these problems. Recently the concept of golden family graphs (HOSOYA,2005) for various sequences of graphs was proposed. Several new sequences of graphs

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were found whose topological indices (HOSOYA, 1971, 1973), Z, are equal to eitherFibonacci or Lucas numbers, or their multiples. They are called golden family graphs,since the ratio Z-values of their consecutive members converges to the golden mean, τ.

The topological index Z is a characteristic quantity obtained by summing the non-adjacent numbers, p(G, k)s, or k-matching numbers, and K(G) is closely related to Z(HOSOYA, 1971, 1973). In this paper it will be shown that the K(G) numbers for severalsequences of polyhex and polyomino graphs are Fibonacci or Lucas numbers, or theirmultiples, and the ratio of K(G)s of their consecutive terms converges to τ.

2. Enumeration Algorithms for K(G)

It is well known that K(G) = 2 for a hexagon corresponding to the pair of the Kekuléstructures of the benzene molecule, or hexagon graph (Fig. 1). By fusing hexagons one byone to benzene, linearly and in a zigzag manner one can obtain two sequences of polyhexgraphs, In, and Wn, as in Fig. 2, where n is the number of hexagons. In chemistry, theycorrespond, respectively, to the carbon atom skeletons of linear polyacenes and zigzagpolyacenes. Their K(G) values are expressed by n + 1 and Fn+1, respectively (GORDON andDAVISON, 1952), as

In = n + 1 (1)

and

Wn = Fn+1, (2)

where Fn is the Fibonacci number defined by

Fig. 1. The Kekulé number, or perfect matching number, of the benzene graph is two.

The Golden Family Graphs of a New Category 407

Fn = Fn–1+ Fn–2 (n ≥ 2) (3)

with

F0 = F1 = 1. (4)

Note that the initial conditions (4) are different from the conventional ones (see VOROBIEV,1961; HOGGATT, 1969). The reason is explained elsewhere (HOSOYA, 2005). The K(G)value of the latter increases far more rapidly than the former, reflecting the relative stabilityof these two sequences of hydrocarbon molecules.

Among a variety of algorithms for enumerating K(G)s of polyhexes, only the onediscovered by GORDON and DAVISON (1952) will be explained here because of its ease in

Fig. 2. Three series of polyhex graphs, In, Wn, and Jn, and their K(G) values. The latter two are the golden familygraphs.

Fig. 3. a) and b): Examples of the algorithm for enumerating the K(G) value of a growing hexagonal animal, oran unbranched catacondensed benzenoid hydrocarbon. Doubly framed and bold hexagons, respectively,indicate the start and goal hexagons. The corner edge drawn with bold line indicates a kink. c) and d):Corresponding diagrams for polyomino graphs.

408 H. HOSOYA

deriving K(G) for a growing hexagonal animal without branching (“unbranchedcatacondensed benzenoid hydrocarbon” in chemical terminology. See Fig. 3).

This kind of animal, or benzenoid chain, has two terminal hexagons, irregardless ofwhich is head or tail. For enumerating its K(G) value one can start from either of them.Figures 3a and b, respectively, illustrate the cases where the left- and right-most hexagonsare chosen as the starting point. First K(G) = 2 is assigned to the starting hexagon. ThenK(G) = 3 is assigned to the adjacent hexagon, because we already know I2 = F2 = 3. Up tohere the procedures in Figs. 3a and b are the same. However, the assignment of K(G) to thethird hexagon differs. In Fig. 3a, I3 = 4 is given according to Eq. (1) since the third hexagonadds linearly, while in Fig. 3b, F4 = 5 is given according to Eq. (2) since the third hexagonadds in a zigzag manner. The fourth hexagon in Fig. 3a grows in a zigzag manner afterpassing a “kink” (bend), and the recursion formula Eq. (3) is applies, i.e., by adding thenumbers assigned to the nearest two ancestors one gets 4 + 3 = 7. This is also the case with

Fig. 4. Smaller catacondensed polyhex series of the golden family graphs, whose K(G) values are eitherFibonacci or Lucas numbers, or their multiples. The dots indicate the start of the series.

The Golden Family Graphs of a New Category 409

the fourth hexagon in Fig. 3b, and we have 5 + 3 = 8. The fifth hexagon in either Figs. 3aor b grows linearly from their last kink. In this case K(G) is the sum of the numbers assignedto the ancestors of the nearest and the one just before the last kink. That is, 7 + 3 = 10 and8 + 3 = 11, respectively. By following these two rules of addition one gets the same valueK(G) = 27 for this polyhex graph irrespective of the choice of starting hexagon. Thisalgorithm can be proved by using a recursion formula explained in detail elsewhere(HOSOYA, 2005). The inclusion-exclusion principle provides the essence of the proof.

By using this algorithm the perfect matching numbers of the sequences of zigzagpolyacene graphs are shown to be the Fibonacci numbers. It is also shown that the K(G)sof the sequences of graphs, Jn, in Fig. 2, are Lucas numbers. Both sequences of graphs maybe called golden family graphs since the ratios of consecutive members of the sequence ofK(G)s converge to τ.

3. Golden Family of Polyhex Graphs

From the database of the K(G) values for a number of polyhex graphs (HOSOYA andYAMAGUCHI, 1975; HOSOYA et al., 1986) one can choose various sequences of goldenfamily graphs. The members of this family rapidly increase as the size of the graphsincreases. Thus in Fig. 4 only those smaller members are shown whose K(G) values areeither Fibonacci or Lucas numbers, or the multiples of the latter, where a filled circleindicates the start of the sequence. Two sequences of graphs, Wn, and Jn, given in Fig. 2,are included in Fig. 4. All the graphs in Fig. 4 are so-called catahexes (TRINAJSTIC, 1992),in which there is no vertex at which three hexagons meet. In other words a catahex is apolyhex whose dual graph is a tree.

As implied by the algorithm in Fig. 3, for the catahexes larger than tetrahexes therearises a possibility for the existence of isomeric members with the same K(G) value as Wn,and the number of these isomers increases quite rapidly with n. Then all the members of thegolden family graphs grow by entangling with their isomers (red, blue, and light blue) asseen in Fig. 4.

Those polyhexes where there is more than one vertex at which three hexagons meet arecalled perihexes. Namely, the dual of a perihex is a non-tree. Although a systematicenumeration of them has not yet been performed, a sequence of golden family graphs whoseK(G) values are the triple of Fn were found as shown in Fig. 5. Thus it is possible that thereexists other golden family members in larger perihexes.

The hydrocarbon molecule corresponding to the second smallest member of this seriesis benzopyrene notoriously famous for its strong carcinogenic property.

Fig. 5. An example of pericondensed polyhex series of the golden family graphs.

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Fig. 7. Smaller catacondensed polyomino series of the golden family graphs, whose K(G) values are eitherFibonacci or Lucas numbers, or their multiples. The dots indicate the start of the series.

Fig. 6. Diagram showing the equivalency of the parfect marching in polyhex and polyomino graphs.

4. Golden Family of Polyomino Graphs

The K(G) value of the linear polyomino, e.g. Ln, with n squares is known to be equalto that of the zigzag polyacene, Wn, with n hexagons. That these two counting problems areessentially the same is explained diagrammatically by Fig. 6. Namely, a given perfectmatching pattern for Wn corresponds to one and only one perfect matching pattern for Ln,and vice versa, meaning that K(Wn) = K(Ln).

Then one can say that the sequence of Ln graphs also belong to the golden familygraphs. The correspondence between polyhexes and polyominoes is not limited to the pair,

The Golden Family Graphs of a New Category 411

Wn and Ln, but is found in other pairs of growing animals. See Figs. 3c and d, where theGordon and Davison algorithm for polyhexes is expanded to the corresponding polyomino.

From the compiled database (MOTOYAMA and HOSOYA, 1976) for polyominoes anumber of sequences of polyomino graphs were found to belong to the golden family asshown in Fig. 7. They are the so-called cataominoes, whose dual graphs are trees.

The linear polyomino graph, Ln, has the largest K(G) value among their isomericpolyomino graphs, while the linear polyacene graphs, In, have the smallest value among theisomeric polyacene graphs. This property can be understood by examining the discussionpertinent to Figs. 3a–d.

Although a systematic study has not yet been performed for “periominoes,” a sequenceof graphs as shown in Fig. 8 were found to belong to the golden family graphs. Either byfusing, branching, or kinking, the K(G) value of polyomino graphs is known to decrease.This property which is opposite to polyhexes can be explained by use of the recursionformulas.

Although the results introduced in this paper have no physico-chemical meaning,these graph-theoretical counterparts of the Fibonacci and Lucas numbers are helpful notonly as a proof technique but also for a global understanding of the mathematical structureof the problems where recursive sequences of numbers play an important role.

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Fig. 8. An example of pericondensed polyomino series of the golden family graphs.

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